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The Optimization Problem for Functions of Bounded Variation

Wan Li, Shuang Mou, Baocheng Zhu

Communications in Mathematics and Statistics, https://doi.org/10.1007/s40304-025-00449-2

For a function of bounded variation f in
$R n$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^n$$\end{document}
, we consider the optimization problem of affine total variation, subject to a constraint on the LYZ body
$⟨ f ⟩$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle f\rangle $$\end{document}
under affine transformations, along with its dual problem. As applications, we also derive properties of the solutions to the related optimization problem, as well as properties of the LYZ body.
Asymptotic Stability for Non-equicontinuous Markov Semigroups

Fuzhou Gong, Yong Liu, Yuan Liu, Ziyu Liu

Communications in Mathematics and Statistics, https://doi.org/10.1007/s40304-025-00447-4

We prove that the asymptotic stability, also known as the weak mixing, is equivalent to a lower bound condition together with the eventual continuity. The latter is a form of weak regularity for Markov–Feller semigroups that generalizes the e-property. Additionally, we provide an example of an asymptotically stable Markov semigroup with essential randomness that does not satisfy the e-property.
Limit Behavior of Deformed Hermitian–Yang–Mills Metrics on Kähler Surfaces

Xiaoli Han, Xishen Jin

Communications in Mathematics and Statistics, https://doi.org/10.1007/s40304-025-00445-6

In this paper, we consider the limit behavior of a sequence of deformed Hermitian–Yang–Mills metrics
$F m$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_m$$\end{document}
on
$L ⊗ m$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{\otimes m}$$\end{document}
where L is an ample line bundle over a Kähler surface
$(X, ω)$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X, \omega )$$\end{document}
. If the cohomology class
$c 1 (L)$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_1(L)$$\end{document}
admits a solution of the J-equation, then we prove that
$F m$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_m$$\end{document}
will converge to it. Furthermore, we also consider a boundary case. In this case, we prove that
$F m$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_m$$\end{document}
will converge to a singular Kähler metric away from a finite number of curves with negative self-intersection on the surface.
On the Robust Nonparametric Regression Estimate in the Single Functional Index Model

Billal Bentata, Said Attaoui, Elias Ould-Saïd

Communications in Mathematics and Statistics, https://doi.org/10.1007/s40304-025-00444-7

In this work, we construct and study a family of robust nonparametric estimators for a regression function based on kernel methods. The data are functional, independent and identically distributed, and are linked to a single-index model. Under general conditions, we establish the pointwise and uniform almost complete convergence, as well as the asymptotic normality of the estimator. We explicitly derive the asymptotic variance and, as a result, provide confidence bands for the theoretical parameter. A simulation study is conducted to illustrate the proposed methodology.
Stability of the Two-Dimensional Point Vortices in Euler Flows

Dengjun Guo

Communications in Mathematics and Statistics, https://doi.org/10.1007/s40304-024-00436-z

We consider the two-dimensional incompressible Euler equation
$∂ t ω + u · ∇ ω = 0, ω (x, 0) = ω 0 (x) .$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}{\left\{ \begin{array}{ll} \partial _t \omega + u\cdot \nabla \omega =0, \\ \omega (x,0)=\omega _0(x). \end{array}\right. }\end{aligned}$$\end{document}
We are interested in the cases when the initial vorticity has the form
$ω 0 = ω 0, ϵ + ω 0 p, ϵ$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _0=\omega _{0,\epsilon }+\omega _{0p,\epsilon }$$\end{document}
, where
$ω 0, ϵ$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _{0,\epsilon }$$\end{document}
is concentrated near M disjoint points
$p m 0$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_m^0$$\end{document}
and
$ω 0 p, ϵ$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _{0p,\epsilon }$$\end{document}
is a small perturbation term. We prove that for such initial vorticities, the solution
$ω (x, t)$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega (x,t)$$\end{document}
admits a decomposition
$ω (x, t) = ω ϵ (x, t) + ω p, ϵ (x, t)$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega (x,t)=\omega _{\epsilon }(x,t)+\omega _{p,\epsilon }(x,t)$$\end{document}
, where
$ω ϵ (x, t)$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _{\epsilon }(x,t)$$\end{document}
remains concentrated near M points
$p m (t)$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_m(t)$$\end{document}
and
$ω p, ϵ (x, t)$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _{p,\epsilon }(x,t)$$\end{document}
remains small for
$t ∈ [0, T]$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \in [0,T]$$\end{document}
. As a consequence of such decomposition, we are able to consider the initial vorticity of the form
$ω 0 (x) = ∑ m = 1 M γ m ϵ 2 η (x - p m 0 ϵ)$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _0(x)=\sum _{m=1}^M \frac{\gamma _m}{\epsilon ^2}\eta (\frac{x-p_m^0}{\epsilon })$$\end{document}
, where we do not assume
$η$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document}
to have compact support. Finally, we prove that if
$p m (t)$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_m(t)$$\end{document}
remains separated for all
$t ∈ [0, + ∞)$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in [0,+\infty )$$\end{document}
, then
$ω (x, t)$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega (x,t)$$\end{document}
remains concentrated near M points at least for
$t ≤ c 0 | log A ϵ |$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \le c_0 |\log A_{\epsilon }|$$\end{document}
, where
$A ϵ$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{\epsilon }$$\end{document}
is small and converges to 0 as
$ϵ → 0$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon \rightarrow 0$$\end{document}
.
Global Existence and Uniqueness of Pathwise Solution to the Stochastic 2D Inviscid Boussinesq Equations

Shijia Zhang, Guoli Zhou

Communications in Mathematics and Statistics, https://doi.org/10.1007/s40304-024-00431-4

Global well-posedness of 2D inviscid Boussinesq equations is unsolved. In the present work, we find that if this inviscid hydrodynamics equation is perturbed by noise, the global well-posedness holds in high probability with initial data satisfies a certain Gevrey-type bound. Moreover, the unique global solution to the stochastic inviscid 2D Boussinesq equation is bounded by the initial data.
A General Class of Transformation Cure Rate Frailty Models for Multivariate Interval-Censored Data

Shuwei Li, Liuquan Sun, Lianming Wang, Wanzhu Tu

Communications in Mathematics and Statistics, https://doi.org/10.1007/s40304-024-00428-z

The promotion time or non-mixture cure model is a popular tool for analyzing failure time data with a cured fraction and its usefulness in survival analysis has been well recognized. Although a number of inference procedures under this model have been proposed for univariate interval-censored data, corresponding estimation methods under multivariate interval censoring are still undeveloped because of the challenges in maximizing the observed data likelihood function with complex form. In this paper, we investigate the inference procedure for a class of generalized promotion time cure models, namely transformation cure rate frailty models, with multivariate interval-censored data. The class of models is quite flexible and general and includes the proportional hazards and proportional odds cure rate frailty models as special cases. An expectation-maximization algorithm is developed to calculate the nonparametric maximum likelihood estimators, and the asymptotic properties of the obtained estimators are derived with the empirical process techniques. Extensive simulation studies demonstrate the reliable and satisfactory empirical performance of the proposed method. It is then applied to a set of sexually transmitted infection data arising from an epidemiological study for illustration.
Testing Equality of Two High-Dimensional Correlation Matrices

Guanghui Cheng, Zhi Liu, Qiang Xiong

Communications in Mathematics and Statistics, https://doi.org/10.1007/s40304-024-00427-0

This paper is concerned with testing the equality of two high-dimensional Pearson correlation matrices without any structural assumption under normal populations. A U-statistic based on the Frobenius norm of the difference between two transformational correlation matrices is proposed for testing the equality of two correlation matrices when both sample sizes and dimension tend to infinity. And the asymptotic normality of the proposed testing statistic is also derived under the null and alternative hypotheses. Moreover, the asymptotic power function is also presented. Simulation studies show that the proposed test performs very well in a wide range of settings and can be allowed for the case of large dimensions and small sample sizes.
A New Class of IMSE-Based Criteria for Optimal Designs in Multi-response Random Coefficient Regression Models

Lei He, Rong-Xian Yue

Communications in Mathematics and Statistics, https://doi.org/10.1007/s40304-024-00426-1

A new class of criteria for optimal designs in random coefficient regression (RCR) models with r responses is presented, which is based on the integrated mean squared error (IMSE) for the prediction of random effects. This class, referred to as
$IMSE r, L$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{IMSE}_{r,L}$$\end{document}
-class of criteria, is invariant with respect to different parameterizations of the model and contains
$IMSE$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{IMSE}$$\end{document}
- and G-optimality as special cases for the prediction in univariate response situations. General equivalence theorems for
$IMSE r, L$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{IMSE}_{r,L}$$\end{document}
-criteria are established for
$L ∈ [1, ∞)$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L\in [1,\infty )$$\end{document}
and
$L = ∞$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L=\infty $$\end{document}
, respectively, which are used to check
$IMSE r, L$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{IMSE}_{r,L}$$\end{document}
-optimality of designs.
$IMSE r, L$
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{IMSE}_{r,L}$$\end{document}
-optimal designs for linear and quadratic bi-response RCR models are given for illustration.
Smoothness of the Optimal Transport Map on Riemannian Products of Spheres

Jian Ye

Communications in Mathematics and Statistics, https://doi.org/10.1007/s40304-024-00423-4

This paper approaches the smoothness of the optimal transport map on Riemannian products manifolds. By classical continuity method, we give an alternative proof of the smoothness of the optimal transport map on Riemannian products of the round spheres. In addition, we also prove that the smoothness of the optimal transport map is not stable in Riemannian products of round spheres.

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